Experimental Study - 福岡工業大学学術機関リポジトリ

We demonstrate the performance of our proposed method by experimental studies. In particular, we here apply the proposed method to a problem on object contour modeling. Now, suppose that we are given a periodic B-spline curvex(t)and the corresponding control polygonsMfor a set of data

sampled from the following periodic functionf(t), f(t) = p

p²(t) +q²(t), (5.16)

with

p(t) = 3 +r(t) cos(2πt/36), q(t) = 3 +r(t) sin(2πt/36),

(5.17) and

r(t) = 2 + sin(10πt/36). (5.18)

Figure 5.1 illustrates the designed periodic curvesx(t). Here, the green dashed line with the black asterisk denotes f(t) with the data. The blue line is the designed periodic B-spline curvex(t). We then construct a sequence of the original control points for periodic spline curves x(t)by using the theory of smoothing splines [41]. Here,α is set asα=10. Hence, the total number of control points M is set asM = 39and the time interval[t₀, t_m]are[0,36]. Using the our proposed method in this

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

Figure 5.1: Designed periodic B-spline curvex(t).

chapter, we design the compact periodic B-spline curvesx(t)¯ from the given curvex(t). Also,∆sis set as∆s = 0.02, and henceN is a set of data. The design examples are illustrated in Figures 5.2-5.10. Here,l_maxin (5.11) is set asl_max = 0,1and2withΛ₀ =I₂,Λ₁ = 10I₂ andΛ₂ = 10²I₂, where I₂ ∈ R^2×2 denotes an identity matrix. Also, for each case of Figures 5.2-5.10 were the result setting asK = 31 ( ¯M = 35), K = 29 ( ¯M = 33), andK = 25 ( ¯M = 29)from original control points. In

Figures 5.2(a)-5.10(a), the given x(t)and the reconstructed x(t)¯ are plotted in the polar coordinate systemO −pq. Here, the red and the blue lines denote the given curves x(t)and the reconstructed compact periodic B-spline curvesx(t). Also, green dashed lines with the black asterisk denote¯ f(t) with data. Also, in Figures 5.2(b)-5.10(b), the weights w_i obtained by solving in (5.14) are plotted in the blue line with square marks. In Figures 5.2(c)-5.10(c) we show that the result of given curve x(t)and the reconstructed curvex(t). Then, the corresponding control polygons¯ M(red dashed lines with circle marks) andM¯ (blue dashed lines with cross marks) are plotted in Figures 5.2(d)-5.10(d), respectively.

From these results, we see that our proposed method relatively works well for the case ofl_max = 2. In the case ofl_max = 0, noting that the control polygon Mrepresent the geometrical outline of curves x(t), we may see that the control points are selected such that a discrepancy betweenMandMˆ by the dynamic programming. Such a selection strategy may often cause that a linear sequence of control points is unselected intensively (see e.g., Figures 5.2, 5.5, and 5.8). Then, the knot point interval corresponding to the unselected control points becomes wider. Hence, representing compact periodic B-spline curves in a class of cubic NURBS may become difficult only by adjusting the weights w_i. Aslmax in (4.11) becomes large, the unselected control points become scattered as shown in Figures 5.3-5.4, 5.6-5.7 and 5.9-5.10. We then may observe that the approximation gets better.

From these results, we observe that the approximations for the case ofl_max = 2 are better than the cases of l_max = 0 and 1, but we cannot avoid that the approximation gets worse as the number of selected control pointsK (orM) becomes small.¯

Also, we here compare the above results with the case where the control points with a specified number of K are selected randomly. Setting K as K = 29 ( ¯M = 33) for the cases of contour model inpq-plane, we repeatedly construct the periodic B-spline curvesx(t)¯ using the method in this Chapter one hundred times and their results are plotted in Figure 5.11. We may see that these results indicate that randomly selecting the dominant control points often leads to the unstable reconstruction of compact periodic curves. Comparing these results with the results in Figure 5.7, it is obvious that our proposed method is more effective than the cases of this random control point selection.

The most remarkable thing is that the same scheme as the case of planar B-spline curves in Chapter 4 works well even for the case with periodic constraints.

0 1 2 3 4 5 6 0

1 2 3 4 5 6

0 5 10 15 20 25 30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a) Contour model of the compact periodic B-spline curvesx(t).¯ (b) Weightw_i.

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

Figure 5.2: Result of Compact periodic B-spline curves whenl_max = 0, K = 31(M¯ = 35).

0 1 2 3 4 5 6 0

1 2 3 4 5 6

0 5 10 15 20 25 30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a) Contour model of the compact periodic B-spline curvesx(t).¯ (b) Weightw_i.

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

Figure 5.3: Result of Compact periodic B-spline curves whenl_max = 1, K = 31(M¯ = 35).

0 1 2 3 4 5 6 0

1 2 3 4 5 6

0 5 10 15 20 25 30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a) Contour model of the compact periodic B-spline curvesx(t).¯ (b) Weightw_i.

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

Figure 5.4: Result of Compact periodic B-spline curves whenl_max = 2, K = 31(M¯ = 35).

0 1 2 3 4 5 6 0

1 2 3 4 5 6

0 5 10 15 20 25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a) Contour model of the compact periodic B-spline curvesx(t).¯ (b) Weightw_i.

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

Figure 5.5: Result of Compact periodic B-spline curves whenl_max = 0, K = 29(M¯ = 33).

0 1 2 3 4 5 6 0

1 2 3 4 5 6

0 5 10 15 20 25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a) Contour model of the compact periodic B-spline curvesx(t).¯ (b) Weightw_i.

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

Figure 5.6: Result of Compact periodic B-spline curves whenl_max = 1, K = 29(M¯ = 33).

0 1 2 3 4 5 6 0

1 2 3 4 5 6

0 5 10 15 20 25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a) Contour model of the compact periodic B-spline curvesx(t).¯ (b) Weightw_i

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

Figure 5.7: Result of Compact periodic B-spline curves whenl_max = 2, K = 29(M¯ = 33).

0 1 2 3 4 5 6 0

1 2 3 4 5 6

0 5 10 15 20 25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a) Contour model of the compact periodic B-spline curvesx(t).¯ (b) Weightw_i.

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

Figure 5.8: Result of Compact periodic B-spline curves whenl_max = 0, K = 25(M¯ = 29).

0 1 2 3 4 5 6 0

1 2 3 4 5 6

0 5 10 15 20 25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a) Contour model of the compact periodic B-spline curvesx(t).¯ (b) Weightw_i.

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

Figure 5.9: Result of Compact periodic B-spline curves whenl_max = 1, K = 25(M¯ = 29).

0 1 2 3 4 5 6 0

1 2 3 4 5 6

0 5 10 15 20 25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a) Contour model of the compact periodic B-spline curvesx(t).¯ (b) Weightwi.

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

0 5 10 15 20 25 30 35 40

1 2 3 4 5 6 7 8

Figure 5.10: Result of Compact periodic B-spline curves whenl_max = 2, K = 25(M¯ = 29).

0 1 2 3 4 5 6

Figure 5.11: Compact periodic B-spline curves x(t)¯ for the case where the control points are randomly selected.

Concluding Remarks

In this thesis, we considered the problem of designing the compact B-spline curves by using only the dominant control points. In particular, we here developed such a design method for the typical two types of B-spline curves “planar B-spline curve” and “periodic B-spline curves”. The central issue was how to optimally select dominant control points. For such an optimal dominant control point selection, we here introduced an optimization approach using a dynamic programming (DP) method.

Namely, it was shown that the selection problem is formulated as a graph problem and is solved by DP. The merit of using the DP approach is to enable us to reduce the amount of computation time unlike the ordinary approaches- such as the trial-and-error approach exhibited in Lyche and Mørken’s work [28]. In addition, what made the paper noteworthy was that a new idea of knots selections based on multi-level error functions is introduced, where the term ‘multi-level’ means that not only function values of a given curve but also its derivatives are considered. Also, we showed that representation of compact B-spline curves using the selected control points can be realized using NURBS (Non-Uniform Rational B-splines) since the allocation of their knot points are generally non-uniform. The methods for the planar and periodic splines were applied for the character design using the so-called dynamic font method and the contour modeling problem for the deformable objects, respectively. The performances were demonstrated by experimental studies.

In Chapter 1, we introduced the background including some related works and the purpose of this thesis as an introduction.

In Chapter 2, we presented the fundamental theory of spline curves. Therein, the polynomial curves and Bezier curves, and their properties were presented. Then, we presented B-spline as well as

Non-76

Uniform B-splines (NURBS), which were used throughout this thesis.

In Chapter 3, we presented the dynamic programming including basics of graph theory, which are mainly used in order to solve the problem of selecting dominant control points.

In Chapters 4 and 5, we considered the problem of designing the compact B-spline curves for the two cases of the planar spline curves and periodic spline curves, respectively. The planar spline curves in Chapter 4 are curves that are represented as a parametric equation of the form(x, y) = (x(t), y(t)) for curves x(t)andy(t). Then, our main task was to design compact B-spline curves independently forX andY directions. In addition, we confirmed such the task was basically similar to the case of periodic spline curves where we need to impose the periodic constraints. For achieving such tasks, we developed the method for selecting the dominant control points by employing the DP approach.

Specifically, the directed acyclic graph (DAG) was constituted from the sequence of control points.

Then, the optimal path was found by DP so that some cost function is minimized, and then the domi-nant control points were obtained as the optimal path corresponding to the control polygon consisting of dominant control points. Regarding such the cost function, we here introduced the so-called “multi-level function” that not only function values of a given curve but also its derivatives are considered. In addition, what is remarkable about the proposed selection method is that the computation complex-ity is quite low because of the DP-based algorithm. In fact, the complexcomplex-ity to find the K-dominant control points fromM¯ isO(KM¯²)[45]. Then, we showed that the representation of compact planar B-spline curves is realizable by introducing NURBS. We demonstrated the performance with some experimental studies. Then, we observed that introducing multi-level error will result in better ap-proximations of compact planar B-spline curves, although it is unavoidable that the apap-proximations get worse as the number of selected control points becomes small. Also, we observed through some experimental studies that random selection of the dominant control points often leads to an unstable reconstruction of compact spline curves. Thus, it was obvious that our proposed method is more effective than cases of random control point selection.

For improving the issues on approximation errors, it remains to develop some method for relocating the knot points corresponding to the selected control points, which is left for a future work. Solving such the knot points’ relocation is generally not easy, and some strategies have been proposed by many researchers. In particular, the strategy which we have to develop here may be based on the idea of “free knots” spline since we could effectively select the dominant control points by our proposed method. Such the relocation method has been developed by Schwetlick, et al. [56]. Therein, the

method consists of minimizing some global criterion over both the position of the knots and the coefficients of the splines. However, what we want to develop may be a method for relocating only the knot points since we want to hold the outline of the control polygon from the viewpoint of user-friendliness on curve design. On the other hand, the advantage of using B-splines is the dimensional extendability. That is, we may extend the results of this thesis to the higher dimensional cases-such as surface and three-dimensional objects, etc. This will be another future work.

[1] C. F. Martin, S. Sun and M. Egerstedt, Optimal Control, Statistics and Path Planning, Mathemat-ical and Computer Modeling, Vol.33, No.1–3, pp.237–253, 2001.

[2] H. Ni, Z. Li and H. Song, Moving Least Square Curve and Surface Fitting with Interpolation Conditions, Proceedings of 2010 International Conference on Computer Application and System Modeling (ICCASM 2010), pp.300–304, Taiyuan, China, Oct.22-24, 2010.

[3] B. W. Silverman, Some Aspects of The Spline Smoothing Approach to Nonparametric Regression Curve Fitting with Discussion, Journal of the Royal Statistical Society, Vol.47, No.1, pp.1–52, 1985.

[4] R. Kress,Numerical Analysis, New York : Springer–Verlag, 1998.

[5] X. Zhang and K. K. Mei, Time-Domain Calculation of Microstrip Components and The Curve-Fitting of Numerical Results, Proceedings of IEEE MTT-S International Microwave Symposium Digest, pp.313–316, Long Beach, CA, USA, Jun.13-15, 1989.

[6] L. J. Wang, W. S. Hsieh, T. K. Truong, I. S. Reed, T. C. Cheng, A Fast Efficient Computation of Cubic-Spline Interpolation in Image Codec,IEEE Trans. Signal Process., Vol.49, No.6, pp.1189–

1197, 2001.

[7] O. Faugeras, Three-Dimensional Computer Vision –A Geometric Viewpoint–, Cambridge, MA : MIT Press, 1996.

[8] N. Ray, A Concave Cost Formulation for Parametric Curve Fitting: Detection of Leukocytes from Intravital Microscopy Images,Proceedings of 2010 IEEE 17th International Conference on Image Processing, pp.53–56, Hong Kong, China, Sep.26-29, 2010.

[9] K. Takayama and H. Kano, A New Approach to Synthesizing Free Motions of Robotic Manipu-lators Based on a Concept of Unit Motions,IEEE Transactions on Systems, Man, and Cybernetics, Vol.25, No.3, pp.453–463, 1995.

[10] Q. Huang, K. Yokoi, S. Kajita, K. Kaneko, H. Arai, N. Koyachi, K. Tanie, Planning Walking Patterns for a Biped Robot,IEEE Transactions on Robotics and Automation, Vol.17, No.3, pp.280–

289, 2001.

[11] W. Khalil and E. Dombre, Modeling, Identification and Control of Robots, London, U.K. : Hermes Penton Ltd., 2002.

[12] J. J. Craig,Introduction to Robotics : Mechanics and Control, MA : Addison-Wesley, 1989.

[13] W. L. Xu, B. L. Ma and S. K. Tso, Curve Fitting Approach to Motion Planning of Nonholo-nomic Chained Systems, Proceedings of 1999 IEEE International Conference on Robotics and Automation, pp.811–816, Detroit, MI, USA, May.10-15, 1999.

[14] I. J. Schoenberg, Contributions to The Problem of Approximation of Equidistant Data by An-alytic Functions. Part A–On the Problem of Smoothing or Graduation. A First Class of AnAn-alytic Approximation formulae,Quarterly of Applied Mathematics, Vol.4, No.2, pp.45-99, 1946.

[15] I. J. Schoenberg, Contributions to The Problem of Approximation of Equidistant Data by Ana-lytic Function: Part B–On The Problem of Osculatory Interpolation. A Second Class of AnaAna-lytic Approximation Formulae,Quarterly of Applied Mathematics, Vol.4, No.2, pp.112–141, 1946.

[16] H. Fujioka and H. Kano, Optimal Smoothing Splines for Detecting Extrema from Observational Data,International Journal of Statistics and Systems, Vol.1, No.2, pp.111–132, 2006.

[17] H. Fujioka and H. Kano, Periodic Smoothing Spline Surface and Its Application to Dynamic Contour Modeling of Wet Material Objects, IEEE Trans. Systems, Man and Cybernetics, Part A, Vol.39, No.1, pp.251-261, 2009.

[18] J. G. Dunham, Optimum Uniform Piecewise Linear Approximation of Planar Curves, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-8, Issue 1, pp.67–75, 1986.

[19] Y. H. Gu, T. Tjahjadi, Efficient Planar Object Tracking and Parameter Estimation Using Com-pactly Represented Cubic B-spline Curves,IEEE Transactions on Systems, Man, and Cybernetics, Vol.29, No.4, pp.358–367, 1999.

[20] P. Bo, G. Luo and K. Wang, A Graph-Based Method for Fitting Planar B-spline Curves with Intersections,Journal of Computational Design and Engineering, Vol.3, Issue 1, pp.14–23, 2016.

[21] A. Morgand, M. Tamaazousti and A. Bartoli, A Geometric Model for Specularity Prediction on Planar Surfaces with Multiple Light Sources, IEEE Transactions on Visualization and Computer Graphics, Vol.24, Issue 5, pp.1691–1704, 2018.

[22] K. M. Soo, M. S. Ryong and L. K. Hee Motion Planning with Planar Geometric Models Curves, Proceedings 1991 IEEE International Conference on Robotics and Automation, pp.1015–1020, Sacramento, CA, USA, Apr.9–11, 1991.

[23] K. Takayama and H. Kano, Dynamic Font : A New Representational Technology,FSTJ, Vol.32, No.2, pp.192–202, 1996

[24] H. Nakata and H. Kano, Generation of Japanese Cursive Sentences using Optimal Smoothing Splines,J.Inform. Process. Soc. Japan, Vol.44, No.1, pp.134–142, 2003.

[25] N. Y. Graham, Smoothing with Periodic Cubic Splines, The Bell System Technical Journal, Vol.62, Issue 1, pp.101–110, 1983.

[26] H. Kano, H. Fujioka, M. Egestedt and C. F. Martin, Optimal Smoothing Spline Curves and Contour Synthesis, Proc. the 16th IFAC World Congress, pp.297-302, Prague, Czech Republic, July 4-8, 2005.

[27] W. XingCe, Z. MingQuan and L. XinYu, An Automatic Matching Method of Object Contour Curves Based on Periodic Curvature Function,2006 IEEE International Conference on Robotics and Biomimetics, pp.1146–1150, Kunming, China, Dec.7–20, 2006

[28] T. Lyche and K. Mørken, Knot Removal for Parametric B-spline Curves and Surfaces,Computer Aided Geometric Design, Vol.4, Issue 3, pp.217–230, 1987.

[29] H. Park and J. H. Lee, B-spline Curve Fitting Using Dominant Point,Proceedings of Interna-tional Conference on ComputaInterna-tional Science ComputaInterna-tional Science–ICCS 2006, Lecture Notes in Computer Science, Vol. 3992, pp.362–366, May.28-31, 2006.

[30] T. Tjahjowidodo, V. T. Dung, and M. L. Han, A Fast Non-Uniform Knots Placement Method for B-Spline Fitting,Proceedings of 2015 IEEE International Conference on Advanced Intelligent Mechatronics (AIM), pp.1490–1495, Busan, South Korea, Jul.7-11, 2015.

[31] H. Kano, H. Fujioka, M. Egerstedt and C. F. Martin. Optimal Smoothing Spline Curves and Contour Synthesis. 2005 IFAC. 16th Triennial World Congress, Prague, Czech Republic, Vol.38, Issue 1, pp.297–302, 2005.

[32] C. de Boor,A Practical Guide to Splines, New York : Springer–Verlag, 1978.

[33] C. de Boor,I. J. Schoenberg Selected Papers, New York : Springer–Verlag, 1988.

[34] K. Ruohonen,Graph Theory, Tampere University of Technology, 2013.

[35] R. Bellman,Dynamic Programming, Princeton University Press, Princeton, NJ, 1957.

[36] S. S. Ray,Graph Theory with Algorithms and its Applications, India : Springer–Verlag, 2013.

[37] A. Amini, T. Weymouth and R. Jain, Using Dynamic Programming for Solving Variational Problems in Vision,IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol.2, No.9, pp.855–867, 1990.

[38] R. Basri, L. Costa, D. Geiger, and D. Jacobs, Determining The Similarity of Deformable Shapes, Vision Research, Vol.38, Issues 15–16, pp.2365–2385, 1998.

[39] P. F. Felzenszwalb and D. P. Huttenlocher, Distance Transforms of Sampled Functions,Theory of Computing (ToC), Vol.8, No.19, pp.415–428, 2004.

[40] T. C. Hu and J. D. Morgenthaler, Dynamic Programming and Graph Optimization Problems, Computers Math. Applic., Vol.27, No.9–10, pp.53–58, 1994.

[41] H. Kano, H. Nakata and C. F. Martin, Optimal Curve Fitting and Smoothing Using Normalized Uniform B-splines : A Tool for Studying Complex Systems,Applied Mathematics and Computa-tion, Vol.169, Issue 1, pp.96–128, 2005.

[42] H. Park and J. H. Lee, B-spline Curve Fitting based on Adaptive Refinement using Dominant points,Computer Aided-Design, Vol.39, pp.439–451, 2007.

[43] R. Soontornvorn and H. Fujioka, Design of Compact Planar B-spline Curves Using DP Control Point Selection,Proceedings of TENCON 2018 IEEE Region 10 Conference, pp.1476–1479, Jeju, Korea, Oct.28–31, 2018.

[44] R. Soontornvorn and H. Fujioka, Design of Compact Planar B-spline Curves using DP Con-trol Point Selection with Multi–level Error Functions- Towards Usability Improvement in Design

of Dynamic Font-based Characters, IEIE Trans. Smart Processing and Computing, Vol.8, No.2, pp.108–120, 2019.

[45] R. Hu and S. M. Watt, Optimization of Point Selection on Digital Ink Curves, Proceedings of 13th International Conference on Frontiers in Handwriting Recognition, pp.525-530, Bari, Italy, Sept.18-20, 2012.

[46] P. Lancaster and M. Tismenetsky,The Theory of Matrices, Academic Press, 1985.

[47] H. Fujioka, H. Kano, H. Nakata and H. Shinoda, Constructing and Reconstructing Characters, Words and Sentences by Synthesizing Writing Motions,IEEE Transactions on Systems, Man, and Cybernetics, Vol.36, No.4, pp.661–670, 2006.

[48] M. Hosaka,Modeling of Curves and Surfaces in CAD/CAM, New York : Springer–Verlag, 1992.

[49] L. Piegl and W. Tiller,The NURBS Book, Heidelberg : Springer–Verlag, 1995.

[50] D. F. Rogers,Introduction to NURBS: with Historical Perspective, Academic Press, 2001.

[51] Y. Mieno, H. Fujioka and H. Kano, Data Compression of Digital-Ink with Pen-Slips Using Multi-level L1 Smoothing Splines, Proceedings of The 2015 IEEE International Conference on Systems, Man, and Cybernetics, pp.1787–1792, Hong Kong, Oct. 9-12, 2015.

[52] G. Guo, T. Jiang, Y. Wang and W. Gao, Recovering Missing Contours for Occluded Object Detection,IEEE Signal Processing Letters, Vol.19, Issue 8, pp.463–466, 2012.

[53] A. Blake and M. Isard,Active Contours, London : Springer–Verlag, 1998.

[54] G. Wahba,Spline Models for Observational Data, Philadelphia, PA : Soc. Ind. SIAM, 1990.

[55] O. Valenzuela, M. Pasadas, I. Rojas, A. Guillen and H. Pomares, Automatic Knot Adjustment for B-spline Smoothing Approximation Using Improved Clustering Algorithm,2013 IEEE Inter-national Conference on Fuzzy Systems (FUZZ-IEEE), Hyderabad, India, July 7-10, 2013.

[56] H. Schwetlick and T. Schutze, Least Squares Approximation by Splines with Free Knots, BIT Numerical Mathematics, Vol.35, No.1, pp.361–384, 1995.

Award

1. 2017 President Award Fukuoka Institute of Technology, October, 2017.

2. 2016 Excellent Presentation Award ofthe IEEE Fukuoka Section Award Winners, March, 2017.

Journal

1. R. Soontornvorn and H. Fujioka, Design of Compact Planar B-spline Curves Using DP Control Point Selection with Multi-Level Error Functions - Towards Usability Improvement in De-sign of Dynamic Font-based Characters,IEIE Transactions on Smart Processing & Computing, Vol.8, No.2, pp.108-120, 2019.

International Conferences

1. R. Soontornvorn, H. Fujioka and H. Kano, DP-based Control Point Selection of Periodic Splines and Its Application to Object Contour Modeling, Extended Abstracts of the 51th ISCIE Inter-national Symposium on Stochastic Systems Theory and Its Applications (SSS’19), pp.23-24, Aizu-Wakamatsu, Fukushima, Japan, Nov. 1-2, 2019.

2. H. Fujioka, R. Soontornvorn and H. Kano, Constructing Compact Cubic B-spline Curves Using DP-based Dominant Control Point Selection, Extended Abstracts of the 50th ISCIE Interna-tional Symposium on Stochastic Systems Theory and Its Applications (SSS ’18), pp.129-130, Kyoto, Japan, Nov. 1-2, 2018.

3. R. Soontornvorn and H. Fujioka, Design of Compact Planar B-spline Curves Using DP Control Point Selection-Towards Usability Improvement in Design of Dynamic Font-based Characters, Proc. the 2018 IEEE Region 10 Conference (TENCON2018), pp.1470-1473, Jeju, Korean, Oct.

28-31, 2018.

4. J. Sawangphol, R. Soontornvorn, H. Fujioka, S. Anraku, N. Miyamoto, T. Kato, H. Kino, A.

Hidaka and H. Kano, Toward an Understanding of Nanosheet Object Motion from Noisy Mi-croscopy Images Using Deep-Learning Approach (invited paper),Abstracts of the 5th Interna-tional Conference on Nanomechanics and Nanocomposites (ICNN5), pp.89, Fukuoka, Japan, Aug. 22-25, 2018.

5. H. Fujioka, R. Soontornvorn and H. Kano, Design of Compact B-spline Curves Using Optimal Control Point Selection, Extended Abstracts of the 49th ISCIE International Symposium on Stochastic Systems Theory and Its Applications (SSS ’17), pp.43-44, Hiroshima, Japan, Nov.

3-4, 2017.

6. R. Soontornvorn, H. Fujioka, V. Chutchavong and K. Janchitrapongvej, Modeling ECG wave-form using optimal smoothing Bezier-Bernstein curves,Proc. the 2017 IEEE Region 10 Con-ference (TENCON2017), pp.1235-1238, Penang, Malaysia, Nov. 5-8, 2017.

Other Conferences

1. R. Soontornvorn and H. Fujioka, DP-based Control Point Selection of Periodic Splines for Compact Contour Modeling,The 72th Joint Conference of Electrical, Electronics and Informa-tion Engineers in Kyushu, pp.117, Fukuoka, Japan, Sept. 27-28, 2019.

2. R. Soontornvorn and H. Fujioka, Constructing Compact Planar B-spline Curves Using Dy-namic Programming-based Control Point Selection, The 71th Joint Conference of Electrical, Electronics and Information Engineers in Kyushu, pp.253, Oita, Japan, Sept. 27-28, 2018.

3. J. Sawangphol, R. Soontornvorn and H. Fujioka, Design of Hairy Brush Characters Using Con-volutional Encoder-Decoder Network,The 71th Joint Conference of Electrical, Electronics and Information Engineers in Kyushu, pp.252, Oita, Japan, Sept. 27-28, 2018.

ドキュメント内福岡工業大学学術機関リポジトリ (ページ 78-100)